6.1 Quadratic Expressions, Rectangles, and Squares. 1. What does the word quadratic refer to? 2. What is the general quadratic expression?
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1 Advanced Algebra Chapter 6 - Note Taking Guidelines Complete each Now try problem in your notes and work the problem 6.1 Quadratic Expressions, Rectangles, and Squares 1. What does the word quadratic refer to? 2. What is the general quadratic expression? 3. What is the general quadratic equation? 4. What is the general quadratic function? 5. ax 2 + bx + c is a quadratic expression in what form? 6. Give three different specific examples of a quadratic expression.
2 7. Is 2x + 5x a quadratic expression? 8. What is the simplest quadratic expression? 9. The product of what two terms results in the simplest quadratic expression? 10. The product of two is a quadratic expression. 11. What types of formulas involve quadratic expressions?
3 12. Study example Now try the following problem in your a. Suppose a rectangular garden 25 feet by 15 feet has a walkway built around it. If the walkway is w feet wide, write the total area, A, of the garden and walkway as a quadratic function of w in standard form. 14. When we square a linear expression, or take a linear expression to the 2 nd power, or multiply a linear expression by itself the result is: 15. The processes of squaring a linear expression is called: 16. Study example Now try the following problem in your a. (2x 5y) 2
4 18. What is one way to think about squaring a binomial that can help you organize the process? 19. Study example Now try the following problem in your a. Find the area of the square with sides of length (a + b) and write your answer in standard form. Draw a picture of the square. 21. Binomial Square Theorem Must memorize 22. Study example Now try the following problem in your a. Simplify 1 2w 2 2 Summarize Section 6.1:
5 6.2 Absolute Value, Square Roots, and Quadratic Equations 1. What is the geometric definition for the absolute value of a number x? 2. What is the piecewise algebraic definition for the absolute value of a number x? 3. Study example 1 4. Now try the following problem in your a. Solve for y: 34 2y = Sketch the graph of f(x) = x and identify the domain and range of the function. Domain: Range:
6 6. How will you need to enter the 5 into your calculator? 7. How should you interpret 16? 8. What notation needs to be used to indicate you want the negative value of the square root of a number? 9. Study Activity #1 10. Absolute Value Square Root Theorem 11. What is the solution to the simplest quadratic equation x 2 = k
7 12. Study example Now try the following problem in your a. Solve 3x 2 = Study example Now try the following problem in your a. A square and a triangle have the same area. The triangle has base 7 cm and altitude 6 cm. What is the length of a side of the square? 16. What is a rational number? 17. What is an irrational number? 18. What kinds of numbers have square roots that are rational numbers? 19. What kinds of number have square roots that are irrational numbers?
8 Summarize Section 6.2:
9 6.3 The Graph-Translation Theorem 1. In an equation y = f(x) what does replacing x with (x h) do to the graph of the function? 2. In an equation y = f(x) what does replacing y with (y k) do to the graph of the function? 3. What translation does the notation T h, k indicate? 4. Graph-Translation Theorem 5. Study example 1 6. Now try the following problem in your a. Find an equation for the image of the graph of y = 5x 2 under the translationt. 2,6 3
10 7. Corollary for translating parabolas 8. Study example 2 9. Now try the following problem in your a. Let y 2 = -2(x + 3) 2 i. What is the value of h? ii. What is the value of k? iii. Write in T notation. b. How is the graph of the function above related to the graph of y = -2x 2? 10. What is the following form for the equation of a parabola called? y k = a(x h) 2
11 11. Using the equation y k = a(x h) 2 a. What is the vertex b. What is the line of symmetry? c. What three things do you know if a > 0? d. What three things do you know if a < 0? 12. Study example Now try the following problem in your 1 2 a. Sketch the graph of y + 3 = x 2 b. Give an equation for the axis of symmetry of the parabola.
12 14. Study example Now try the following problem in your a. If parabola P has equation y = -2x 2, what is the equation for parabola Q? Summarize Section 6.3:
13 6.4 Graphing y = ax 2 + bx + c 1. What are the two forms for the equation of a parabola?(a quadratic equation) 2. Study example 1 3. Now try the following problem in your a. Are the following equations equivalent? y = 3(x + 1) 2 4 and y = 3x 2 + 6x 1 4. When comparing the two forms below: y k = a(x h) 2 to y = ax 2 + bx + c a in standard form will be the same as the a in vertex form b in standard form will be equal to -2ah from vertex form c in standard form will be equal to ah 2 + k from vertex form a in vertex form will be the same as the a in standard form h in vertex form will be equal to standard form b 2a k in vertex form will be equal to c - standard form from 2 b from 4a
14 5. Theorem dealing with parabola congruency 6. Given the equation of a parabola f(x) = ax 2 + bx + c What is the y-intercept? What is the domain? What is the range? 7. Given the equation of a parabola y k = a(x h) 2 What is the vertex? What is the axis of symmetry? What is the domain? What is the range?
15 8. In the equation for free falling objects 1 2 h = gt + v t + 2 o h o a. What does h represent? b. What does g represent? c. What does vo represent? d. What does ho represent? e. What does t represent?
16 9. Study example Study example Now try the following problem in your a. Suppose a toy rocket is launched so that its height h in meters after t seconds is given by h 2 = 4.9t + 20t i. How high is the rocket after 1 second? ii. How high is the rocket when launched? iii. How high is the rocket after 12 seconds? iv. When is the rocket 13 meters in the air? v. When will the rocket hit the ground? Summarize Section 6.4:
17 6.5 Completing the Square 1. Converting from vertex form to standard form involves expanding the square of the binomial, distributing a, and solving for y. y k = a(x h) 2 to y = ax 2 + bx + c (Vertex form) (Standard form) What method does converting from standard form to vertex form use? y = ax 2 + bx + c to y k = a(x h) 2 (Standard form) (Vertex form) 2. What is a perfect-square trinomial? 3. Draw a diagram of (x + h) 2 = x 2 + 2hx + h 2 4. Study example 1 5. Now try the following problem in your a. What number should be added to x 2 + 5x to make a perfect-square trinomial. Draw a picture to represent x 2 + 5x.
18 6. Theorem about completing the square. 7. Study example 2 8. Now try the following problem in your a. Rewrite the equation below in vertex form: i. y = x x + 90 ii. y = x 2 11x + 4
19 9. Study example Now try the following problem in your a. Rewrite the equation below in vertex form: i. y = 3x 2 12x + 1 ii. y = -5x 2 + 4x Now try the following problem in your a. Suppose a ball is thrown straight up from a height of 4 feet with an initial velocity of 50 feet per second. What is the maximum height of the ball?
20 Summarize Section 6.5:
21 6.6 Fitting a Quadratic Model to Data 1. How many data point do you need in order to fit a quadratic model to data? 2. Study example 1 3. Now try the following problem in your a. Try to create a quadratic model for the data from example 1 using your calculator. 4. Study example 2 5. Now try the following problem in your a. Try to create a quadratic model for the data from example 2 using your calculator.
22 6. Try the following problem in your Jeremy was in charge of scheduling for the local Little League. Each team played each other team twice. He needed to know the total number of games played so that he could provide umpires. The first year, with 4 teams, he scheduled 12 games. The second year, with 5 teams, he scheduled 20 games. The third year with 6 teams, he scheduled 30 games. Find the number of games needed for a league with a. 2 teams. b. 3 teams. c. With the number of teams as the independent variable, make a scatterplot of these data. d. Fit an appropriate model to these data by hand. e. How many games would be necessary for a 10-team league? Summarize Section 6.6:
23 6.7 & 6.10 The Quadratic Formula 1. Quadratic Formula Theorem 2. What is the discriminant of the quadratic equation y = ax 2 + bx + c? (page 402) 3. Draw a graph and identify the value of the discriminant for all possible outcomes when solving a quadratic equation. (page 403) 4. Discriminant Theorem: (page 403)
24 5. Study example 1 6. Now try the following problem in your a. Determine the number of roots and the type of numbers of the roots of the following equations. Then solve the equations. i. 10x 2 13x 3 = 0 ii. 3x 2 + 3x + 8 = 0 iii. 15x 2 + 2x 1 = 0 iv. 16x 2 72x + 81 = 0 v. x 2 + 7x + 1 = 0
25 7. Study example 2 8. Now try the following problem in your a. Use the Pop Fligh s situation below to find out when the ball is 50 feet high. 9. Study example Now try the following problem in your a. The sides of a right triangle are consecutive even integers. What are they? (Hint: Let the length of one side be 2n.) 11. Now try the following problem in your a. The minimum distance d in feet it takes for a certain car to stop is approximated by the formula d = 0.042s s + 4 where s is the speed in miles per hour. If a car took 200 feet to stop, about how fast was it traveling?
26 12. Now try the following problem: A ball is thrown from a height of 5' with a vertical velocity of 40' / sec. Will this ball ever reach a height of 50'? 13. Now try the following problem: Does the graph of y = 6x 2 + 5x 2 have any x- intercepts? 14. Now try the following problem: Does 10x 2 x 3 = 0 have any rational solutions? If so, find them. Summarize Section 6.7 & 6.10:
27 6.8 Imaginary Numbers 1. Solve the following equations showing all the steps: a. t 2 = 400 b. t 2 = Definition of the square root of negative numbers: 3. What type of number are the solutions to 1b. above? 4. Definition of i?
28 5. Theorem for the square root of negative numbers: 6. Study example 1 7. Now try the following problem: a. Solve 2x = 0 8. Study example 2 9. Now try the following problem: a. Show that i 7 is a square root of -7.
29 10. Study example Now try the following problem: a. Simplify i. (8i) 2 ii iii iv Study example Now try the following problem: a. Simplify Summarize Section 6.8:
30
31 6.9 Complex Numbers 1. How are complex numbers formed? 2. Definition of a complex number: 3. When are complex numbers equal? 4. Postulate for complex numbers: 5. How do we add or subtract complex numbers? 6. Study example 1 7. Now try the following problem: a. Add or subtract and simplify. i. (6 5i) + (3 + 4i) ii. (2 + i) (7 2i)
32 8. How do we multiply complex numbers? 9. Study example Now try the following problem: a. Simplify 3i(12 3i) 11. When working with complex numbers, results should always be written in what form? 12. Simplify: a. a. i i 2 b. i 3 c. i 4 d. i Study example Now try the following problem: a. Multiply or divide and simplify: i. (5 + 9i)(2 7i) ii. (1 + i)(1 i)
33 15. Study example Now try the following problem: a. The formula V = ZI relates voltage (V) in volts to current (I) in amps and Impedance (Z) in ohms in an electrical circuit when these quantities are expressed as complex numbers. i. Find V if I = i amps and z = 7-3i ohms 17. What is the complex conjugate of a + bi? 18. When multiplying complex conjugates the result always equals what expression?
34 19. How do we handle dividing by non-real complex numbers? 20. Study example Now try the following problem: 5 + 3i a. Write in a + bi form. 2 + i 22. Create the hierarchy of your current number system. Summarize Section 6.9:
35 6.10 Analyzing Solutions to Quadratic Equations 1. Solving quadratic equations led to what types of numbers? 2. What are the two graphical ways to solve the equation x 2 3x = 10? 3. What is the advantage of first converting the equation to standard form? 4. What is the discriminant of the quadratic equation y = ax 2 + bx + c? 5. Draw a graph and identify the value of the discriminant for all possible outcomes when solving a quadratic equation. 6. Discriminant Theorem:
36 7. What is another name given to solutions of a quadratic equation? 8. Study example 1 9. Now try the following problem: a. Determine the nature of roots of the following equations. Then solve the equations. i. 3x 2 + 3x + 8 = 0 ii. 15x 2 + 2x 1 = 0 iii. 16x 2 72x + 81 = 0 iv. x 2 + 7x + 1 = 0
37 10. Study example Now try the following problem: b. A ball is thrown from a height of 5' with a vertical velocity of 40' / sec. Will this ball ever reach a height of 50'? 12. Now try the following problem: c. Does the graph of y = 6x 2 + 5x 2 have any x-intercepts? 13. Now try the following problem: d. Does 10x 2 x 3 = 0 have any rational solutions? If so, find them. Summarize Section 6.10:
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